# Locally convex space which is not Banach.

I know that all Banach spaces are (Hausdorff)locally convex spaces. I would like to verify that the converse is not true by giving an example of a space which is locally convex but not Banach. I am looking at the linear space $C([0,1],\left\|\cdot\right\|_1)$ of complex-valued functions defined on the compact interval $[0,1]$, where the norm is given by $$\left\|f\right\|_1=\int_{0}^{1}|f(t)|dt$$ for any $f\in C[0,1]$. The normed space $C([0,1],\left\|\cdot\right\|_1)$ is known to be not Banach. I don't have any idea if the said space is a locally convex space. Any tips? Thanks in advance...

• Every normed space is Hausdorff and locally convex. A Banach space is a complete normed space. If you are looking at normed spaces, to be "non-Banach" is the same as to be incomplete. – Willie Wong Feb 1 '13 at 14:01
• Maybe you can add the definition of local convexity you are trying to verify and add some thoughts (or a proof) why Willie's first sentence is correct. – Martin Feb 1 '13 at 14:18